CHE-23

McCabe–Thiele operating line

y = [R/(R+1)] x + xD/(R+1). Rectifying-section material balance.

Reading speed

DistillationMcCabe–Thiele

Governing equation

y=\dfrac{R}{R+1}x+\dfrac{x_D}{R+1}

$x \in \left[0.01, 0.99\right]$

where

R: Reflux ratio (—)
x_D: Distillate mole frac. (—)
x: Liquid mole frac. (—)
y: Vapour mole frac. (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

y=\dfrac{R}{R+1}x+\dfrac{x_D}{R+1}

$x \in \left[0.01, 0.99\right]$

$x = 0.01$ $y = 0.279$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.01,\quad x_{\max} = 0.99

$x \in \left[0.01, 0.99\right]$

$x = 0.01$ $x = 0.01$

Along the curve

③

Response curve — green

y=\dfrac{R}{R+1}x+\dfrac{x_D}{R+1}

$x \in \left[0.01, 0.99\right]$

$x = 0.01$ $y = 0.279$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}y}{\mathrm{d}x}

$x \in \left[0.01, 0.99\right]$

$x = 0.016$ $y^{\prime} = 0.714$

Along the curve

Lecture brief

Historical brief

From CSTR/PFR mole balances and Arrhenius rates to McCabe–Thiele stages and NTU exchangers, chemical engineering is conservation plus equilibrium. The lab is that design arithmetic. This sheet (CHE-23 — McCabe–Thiele operating line) is the form associated with McCabe–Thiele. Working symbols: $R$ , $x_D$ , $x$ $\rightarrow$ $y$ . McCabe and Thiele stepped equilibrium against a straight operating line from a constant-molal-overflow balance.

Purpose

Purpose: compute

y

from

R

x_D

x

in Chemical engineering via

y=\dfrac{R}{R+1}x+\dfrac{x_D}{R+1}

y = [R/(R+1)] x + xD/(R+1). Rectifying-section material balance. Use it when a real chemical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 2.500\,\mathrm{—}

x_D = 0.950\,\mathrm{—}

x = 0.500\,\mathrm{—}

, the governing relation

y=\dfrac{R}{R+1}x+\dfrac{x_D}{R+1}

yields

y = 0.629\,\mathrm{—}

. An x–y diagram, a diagonal, an operating line. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Reflux ratio R2.5 —Distillate mole frac. x_D0.95 —Liquid mole frac. x0.5 —

Outputs

Vapour mole frac. y0.629 —

Reading speed

Watch on YouTube

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Narration of this film

An x–y diagram, a diagonal, an operating line.

McCabe and Thiele stepped equilibrium against a straight operating line from a constant-molal-overflow balance.

Reading speed

Watch on YouTube